Question: Solve for $x$ : $ 4|x - 3| - 2 = 6|x - 3| + 3 $
Answer: Subtract $ {4|x - 3|} $ from both sides: $ \begin{eqnarray} 4|x - 3| - 2 &=& 6|x - 3| + 3 \\ \\ {- 4|x - 3|} && {- 4|x - 3|} \\ \\ -2 &=& 2|x - 3| + 3 \end{eqnarray} $ Subtract $3$ from both sides: $ \begin{eqnarray} -2 &=& 2|x - 3| + 3 \\ \\ {- 3} && {- 3} \\ \\ -5 &=& 2|x - 3| \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{-5} {{2}} = \dfrac{2|x - 3|} {{2}} $ Simplify: $ -\dfrac{5}{2} = |x - 3| $ The absolute value cannot be negative. Therefore, there is no solution.